DUAL SERIES EQUATIONS INVOLVING GENERALIZED LAGUERRE POLYNOMIALS
B. M. SINGH, J. ROKNE, AND R. S. DHALIWAL Received 1 April 2005
An exact solution is obtained for the dual series equations involving generalized Laguerre polynomials.
1. Introduction
We consider the following dual series equations:
∞ n=0
AnL(α)n
(x+b)h
Γ(α+n+ 1) =f(x), 0< x < a, (1.1) ∞
n=0
AnL(σ)n
(x+b)h
Γ(α+n+β) =g(x), a < x <∞, (1.2) whereα+β+ 1> β >1−m,σ+ 1> α+β >0,m is a positive integer, and 0< h <∞, 0≤b <∞, andhandbare finite constants.L(α)n [(x+b)h] is a Laguerre polynomial,An are unknown coefficients, and f(x) andg(x) are prescribed functions.
Srivastava [5,6] has solved the following dual series equations:
∞ n=0
AnL(α)n (x)
Γ(α+n+ 1)=f(x), 0< x < a, (1.3) ∞
n=0
AnL(σ)n (x)
Γ(α+n+β)=g(x), a < x <∞. (1.4) The triple series equations (1.3) and (1.4) are a special case of the dual series equations (1.1) and (1.2) when
h=1, b=0. (1.5)
Recently, Lowndes and Srivastava [3] have solved the triple series equations involving Laguerre polynomials. References for the solutions of dual and triple series equations
Copyright©2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:7 (2005) 1135–1139 DOI:10.1155/IJMMS.2005.1135
involving Laguerre polynomials are given in [3]. Connected to this work, references and solutions for dual series equations are given by Sneddon [4].
The dual series equations (1.1) and (1.2) are new in the literature and have importance due to the closed-form solution. The results of this note are shown to be in agreement with those of Srivastava [5]. The analysis is purely formal and no justification had been given for the change of the order of integrations and summation.
2. Some useful results
In this section, we will discuss some results which are useful in solving dual series equa- tions (1.1) and (1.2). The orthogonality relation for Laguerre polynomials is given by [2, page 292, equation (2)] and [2, page 293, equation (2)], from which we have
∞
0 xαe−xL(α)n (x)L(α)m (x)dx=Γ(α+n+ 1)
Γ(n+ 1) δnm, α >−1, (2.1) whereδnmis the Kronecker delta.
We can easily find, with the help of integrals [2, page 293, equation (5)] and [2, page 405, equation (20)], that
ξ
0xα(ξ−x)β+m−2L(α)n (x)dx
=Γ(α+n+ 1)Γ(β+m−1)
Γ(α+β+m+n) ξα+β+m−1L(α+β+mn −1)(ξ), α >−1,β+m >1,
(2.2) ∞
ξ e−x(x−ξ)σ−α−βL(σ)n (x)dx=Γ(σ−α−β+ 1)e−ξLα+βn −1(ξ), σ+ 1> α+β >0. (2.3) From [1, page 190, equation (27)], we find that
dm dxm
xα+mL(α+m)n (x)=Γ(α+m+n+ 1)
Γ(α+n+ 1) xαLαn(x). (2.4) 3. Solution of dual series equations (1.1) and (1.2)
We assume that
x+b=X1/h, fX1/h−b= f1(X),
gX1/h−b=g1(X), bh=c, (a+b)h=d, (3.1) then the dual series equations (1.1) and (1.2) can be written in the following form:
∞ n=0
AnL(α)n (X)
Γ(α+n+ 1)= f1(X), c < X < d, (3.2) ∞
n=0
AnL(σ)n (X)
Γ(α+β+n)=g1(X), d < X <∞. (3.3)
We assume that
∞ n=0
AnL(α)n (X)
Γ(α+n+ 1)=f2(X), 0< X < c. (3.4) Combining the series equations (3.2) and (3.4), we can write the dual series equations (3.2) and (3.3) in the form
∞ n=0
AnL(α)n (X)
Γ(α+n+ 1)=F(X), 0< X < d, (3.5) ∞
n=0
AnL(σ)n (X)
Γ(α+β+n)=g1(X), d < X <∞, (3.6) where
F(X)=
f2(X), 0< X < c,
f1(X), c < X < d. (3.7)
Multiplying (3.5) byXα(ξ−X)β+m−2, wheremis a positive integer, integrating with re- spect toXover (0,ξ), and interchanging the order of integrations, we find on using (2.2) that
∞ n=0
AnL(α+β+mn −1)(ξ) Γ(α+β+m+n)=
ξ−α−β−m+1 Γ(β+m−1)
ξ
0Xα(ξ−X)β+m−2F(X)dX, 0< ξ < d, (3.8) where
α >−1, β+m >1. (3.9)
If we now multiply (3.8) byξα+β+m−1, differentiate both sidesmtimes with respect toξ, and use formula (2.4), we find that
∞ n=0
AnL(α+βn −1)(ξ) Γ(α+β+n) =
ξ−α−β+1 Γ(β+m−1)
dm dξm
ξ
0Xα(ξ−X)β+m−2F(X)dX, 0< ξ < d, (3.10) where
α >−1, β+m >1. (3.11)
Multiplying (3.6) bye−X(X−ξ)σ−α−β, integrating with respect toxover (ξ,∞), and in- terchanging the order of integrations, we find by using formula (2.3) that
∞ n=0
AnL(α+βn −1)(ξ) Γ(α+β+n) =
eξ Γ(σ−α−β+ 1)
∞
ξ e−X(X−ξ)σ−α−βg1(X)dX, d < ξ <∞, (3.12) where
σ+ 1> α+β >0. (3.13)
The left-hand sides of (3.10) and (3.12) are now identical. Making use of the orthogo- nality relation (2.1), we find from (3.10) and (3.12) that
An=Γ(n+ 1) d
0
e−ξL(α+βn −1)(ξ)F1(ξ)dξ Γ(β+m−1) +
∞
d
ξα+β−1L(α+βn −1)(ξ)G(ξ)dξ Γ(σ−α−β+ 1)
, (3.14) where
F1(ξ)= dm dξm
ξ
0Xα(ξ−X)β+m−2F(X)dX, (3.15) G(X)=
∞
ξ e−X(X−ξ)σ−α−βg1(X)dX, (3.16) provided thatα+β+ 1>1−mandσ+ 1> α+β >0.
With the help of (3.7), (3.15) can be written in the form:
F1(ξ)= dm dξm
c
0Xα(ξ−X)β+m−2f2(X)dX+ ξ
XXα(ξ−X)β+m−2f1(X)dX
, c < ξ.
(3.17) When we put
b=0, h=1, f2(X)=0 (3.18)
in the solution of the dual series equations (1.1) and (1.2), we then obtain the solution of the dual series equations (1.3) and (1.4) and the results are in complete agreement with those of [5].
References
[1] A. Erd´elyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,Higher Transcendental Functions.
Vols. I, II, McGraw-Hill, New York, 1953.
[2] ,Tables of Integral Transforms. Vol. II, McGraw-Hill, New York, 1954.
[3] J. S. Lowndes and H. M. Srivastava,Some triple series and triple integral equations, J. Math. Anal.
Appl.150(1990), no. 1, 181–187.
[4] I. N. Sneddon,Mixed Boundary Value Problems in Potential Theory, North-Holland, Amster- dam, 1966.
[5] H. M. Srivastava,A note on certain dual series equations involving Laguerre polynomials, Pacific J. Math.30(1969), 525–527.
[6] ,Dual series relations involving generalized Laguerre polynomials, J. Math. Anal. Appl.
31(1970), 587–594.
B. M. Singh: Department of Mathematics and Statistics, The University of Calgary, Calgary, AB, Canada T2N 1N4
J. Rokne: Department of Computer Science, The University of Calgary, Calgary, AB, Canada T2N 1N4
E-mail address:[email protected]
R. S. Dhaliwal: Department of Mathematics and Statistics, The University of Calgary, Calgary, AB, Canada T2N 1N4
E-mail addresses:[email protected]; [email protected]
